1. Introduction: Numbers That Refuse to Fit
You already know rational numbers. A rational number is any number you can write as p/q, where p and q are integers and q is not zero.
So 3 is rational — write it as 3/1. The number 0.75 is rational — write it as 3/4. Even -7 is rational — write it as -7/1. And 0.333... (repeating) is rational — that is 1/3.
Here is the question. Can every number be written as p/q?
Think about the number whose square is 2. You learned about this in Class 9. If you put it on a calculator, you see 1.41421356... and the digits keep going, with no repeating block.
Stop scrolling. Try to write sqrt(2) as a fraction p/q. Can you do it? Try for 30 seconds before reading on.
You cannot. And it is not because you are not clever enough — it is because no such fraction exists. That is the whole point of this lesson.
You can now recall that some numbers sit outside the p/q world.